In NZ, (which is probably fairly typical) tower controllers are permitted to identify an aircraft on radar by correlating it to a pilot report reference a position displayed on the radar, by assigning a transponder code and seeing that code appear, by observing the "ident" feature, by observing a full datablock (flight details, or "correlated" to a flight plan), by reference to a distance/bearing of a navaid displayed on the radar map, and by visually correlating an aircrafts position with that displayed on radar.
The method we do not use that centre radar controllers can is visual observation of a turn on radar. That one isn't used much, these days, anyway.Almost everyone has a transponder.
There are other conditions with the use of these methods, but thats the basics.